Investigating Young Group Double Cosets with Computer Algebra

Bill Pletsch
Technical Vocational Institute, Albuquerque, New Mexico, USA

Let us consider a deuteron colliding with another deuteron ignoring charge and
spin.  In the case where after the collision two deuterons are returned there
are only two possible reactions.  Either nothing happened or a particle was
exchanged.  Generalizing this simple problem from scattering theory results in
an excursion into Polya's theory of counting and the theory of double cosets.

Until the advent of computer algebra, the theory of double cosets has been
restricted to a few elegant but computationally impossible theorems.
Impossible in the sense that in principle the calculation can be done but it
will take ten thousand years.

Today, using Computer Algebra much can be calculated quickly.  Using Macsyma
and Maple in the special case of Young group generated double cosets, we will
see just how valuable Computer Algebra can be.  Some surprising and stimulating
patterns emerge after a just few computer algebra experiments.